Measuring selective constraint on fertility in human life histories

Significance

Trade-offs, or constraints, play a substantial role in shaping the ability of organisms to respond adaptively to selection. We derive a simple means for measuring constraints in demographic rates like fertility and survival that builds on commonly used evolutionary demographic measures. We apply this measure to a hypothesized trade-off between fertility and infant survival and show that a 1% reduction in infant mortality would have to be balanced by an approximately 20% increase in fertility to be favored by selection. Our paper should have an important impact on shaping future empirical studies of human life histories and parental investment as it identifies key measures that we expect should be quite variable with respect to ecology, technology, and social systems.

Abstract

Human life histories combine late age at first reproduction, long reproductive span, relatively high fertility, and substantial postreproductive survival. However, even among the most fecund populations, human fertility falls far below its theoretical maximum. The extent of parental care required for successful offspring recruitment and widespread fertility decline under proper economic conditions suggest that selection on fertility is constrained by trade-offs with recruitment. Here we measure the trade-offs between life history traits under selection by approximating the slope of the selective constraint curve on two traits at the observed values. Using a selection of populations that span human demographic space, we find that the substitution elasticity of fertility for infant survival shows age-related patterns, with minimum substitution elasticities ranging from 14 to 22 for the four populations. The age of this minimum occurs earlier in the high-mortality populations relative to generation time than it does in the low-mortality populations. The human curves are qualitatively similar to one of two comparable nonhuman primate age-specific substitution elasticity curves. The curve for rhesus macaques has a similar shape but is shifted down, meaning that the threshold for switching from investing in survival to fertility is lower at all ages. The magnitude of the substitution elasticities is similar between chimpanzees and humans but the shape is quite different, rising more slowly for a longer fraction of the chimpanzee life cycle. The steeply rising substitution elasticities with age in humans has clear implications for the evolution of reproductive senescence.

Human life histories are characterized by late age at first reproduction, long lifespan, and an extended period of low-reproductive-effort fertility spanning 25 y or more (1). In addition, the human life cycle is characterized by a period of extended childhood in which parents invest heavily in the child’s human capital (2, 3). In an age-structured population, fitness is given by the unique real root to the Euler−Lotka equation, written in discrete form (4) as

$$1={\displaystyle \sum _{i=\alpha }^{\beta }{\lambda }^{-i}\left({\displaystyle \prod _{j=1}^{i-1}{P}_j}\right)F_i},$$ [1]

where α is age at first reproduction, β is age at last reproduction, $P_j$ is the age-specific survival probability for age j, $F_i$ is the fertility rate at i, and λ is the multiplicative rate of increase. Fitness is given by λ, which is defined implicitly by Eq. 1.

Since Lewontin’s pioneering paper on the evolution of colonizing ability, attention has focused on age at first reproduction as a central trait for understanding the evolution of life histories (5). For example, in Charnov’s model for female mammals (6), age at first breeding is the control parameter for life histories. Assuming a stationary population, the Charnov model maximizes $R_0$ with respect to α. In a nonstationary population, where fitness is defined by Eq. 1, simple maximization with respect to α suggests that the earliest achievable age at first reproduction should be optimal. However, delayed age at first reproduction appears to be a derived trait in humans (7). Furthermore, although α has been the dominant life history parameter in many models of mammalian life history evolution, there are other means of maximizing fitness when α is constrained.

In much of life history theory, mortality is considered external to the organism, environmentally determined, and therefore largely out of control of the organism (6, 7). All things being equal, we should expect fertility to be maximized in the absence of trade-offs. Of course, some mortality is within the control of the organism if a fraction of the mortality is attributable to reproductive effort (8). To the extent that effort-attributable mortality trades off with fertility, the age-specific schedules of mortality and fertility are expected to represent a compromise between maximizing fertility while minimizing mortality trade-offs due to high reproductive effort.

An effort-mediated trade-off of particular interest is between fertility and immature mortality. Given the long, slow nature of human life histories and the extensive, obligate parental investment required for successful recruitment of human offspring, we are particularly interested in the trade-offs between early survival and fertility. The logic of this investigation follows the same reasoning that underlies the life history model of Charnov (6). Namely, the mortality and growth rates of immatures—and hence their recruitment success—is at least partially under the control of the mother in mammals, and potentially both parents where there is substantial male parental investment. This particular constraint lies at the heart of the classic trade-off between offspring quality and quantity (9), a theme that has been frequently explored in evolutionary anthropology (10–13). Several factors, in particular, suggest that the trade-off between fertility and immature mortality will be especially important. First, the highest force of selection in the human life cycle falls on prereproductive survival (14), suggesting that fitness components that trade off with early survival may be steep. Second, there is an extensive literature on the impact on short birth intervals on the survival of both infants and previous children (12, 15). Third, increased fertility and secondary altriciality of human infants are derived traits essential to understanding the evolution of the human life cycle (1).

The various terms in Eq. 1 will typically make differential contributions to fitness, and changes in some life cycle transitions have a large impact on mean fitness, whereas fitness is more indifferent to changes in others (14). When measured on a proportional scale, such differences in the response of fitness to changes in life cycle transitions are known as fitness elasticities (4). The way that the life cycle structures trade-offs between fertility and mortality is likely to be related to the fitness elasticities of the different life cycle elements. This observation suggests that the relative magnitude of elasticities of different life history traits will be a measure of the life history trade-offs. In this paper, we present a method for measuring the constraints on selection imposed by trade-offs between life cycle elements and apply this to the problem of understanding the fundamental trade-off between fertility and juvenile survival in human life histories.

Methods

Matrix population models describe life history transitions in terms of the rates $a_{ij}$ in which an individual of stage j at time t produces an individual in stage i at time $t+1$ (4). For example, in an age-classified model, if j and i are consecutive age classes, $a_{ij}$ represents an age-specific survival probability, whereas if j is an adult age class and i is the first age class, $a_{ij}$ represents the fertility of age j individuals. The dynamics of a population are characterized by the matrix $\mathbf{A}$ containing all of the transitions $\left\{a_{ij}\right\}$. If $\mathbf{A}$ is irreducible and aperiodic, it will grow asymptotically at a rate λ, which is the dominant eigenvalue of the matrix $\mathbf{A}$. Furthermore, it will have stable age distribution $\mathbf{u}$ and reproductive values $\mathbf{v}$, the dominant right and left eigenvectors of $\mathbf{A}$, respectively.

In the absence of trade-offs, increasing any of the $a_{ij}$ will clearly increase λ. The amount by which λ increases given a small increase in element $a_{ij}$ is known as an “eigenvalue sensitivity” and is given by $s_{ij}=\partial \lambda /\partial a_{ij}=v_i{u}_j$, where $\langle \mathbf{v},\mathbf{u}\rangle =1$. The proportional sensitivity or elasticity of the dominant eigenvalue gives the amount that λ will change on a log scale: $e_{ij}=\partial \log \lambda /\partial \log a_{ij}=\left(a_{ij}/\lambda \right)\left(\partial \lambda /\partial a_{ij}\right)$. Elasticities are proportional sensitivities. For example, if $a_{ij}$ changes by 10%, by what percentage will λ change?

The fitness of a phenotype with slightly different transition rates $a_{ij}+D_{ij}=a_{ij}\left(1+d_{ij}\right)$ will be $\lambda +\mathrm{\Delta }\lambda$ and selection will favor phenotypes for which

$$\mathrm{\Delta }\lambda ={\displaystyle \sum _{ij}{s}_{ij}{D}_{ij}}={\displaystyle \sum _{ij}{e}_{ij}{d}_{ij}>0}.$$

In general, a constraint can be written as a functional relationship of the form $f\left(\log h,\log g\right)=0$. Assume that the two rates, g and h, are near their respective optima. Let $e_g=\partial \log \lambda /\partial \log g$. Now change these rates to $g\left(1+d_g\right)$ and $h\left(1+d_h\right)$. Because the rates are near their optima, the change in fitness that follows from a change in the vital rates is nearly zero,

$$\mathrm{\Delta }\lambda =e_g{d}_g+e_h{d}_h\approx 0.$$ [2]

Although the rates have changed, the constraint should nonetheless hold. Because the logarithms of g and h change by $d_g$ and $d_h$, the change in the constraint function is

$$\frac{\partial f}{\partial \log g}\hspace{0.17em}{d}_g+\frac{\partial f}{\partial \log h}\hspace{0.17em}{d}_h=0.$$ [3]

Let $f_g=\partial f/\partial \log g$ be the partial derivative of $f\left(\log g,\log h\right)$ with respect to its first argument and $f_h=\partial f/\partial \log h$ be the analogous quantity for the second argument. Both Eqs. 2 and 3 can be solved for $\left(d_g/d_h\right)$, which means that

$$\frac{e_g}{e_h}=\frac{f_g}{f_h}.$$ [4]

Therefore, the ratio of elasticities of the elements of g and h equals the slope of the constraint that connects them (Fig. 1). A similar result was derived by Caswell (16) for the sensitivities of the projection matrix and used to explore the evolution of senescence. One interpretation of this ratio is that it is the required return on fitness for exchanging a one-unit proportional investment in h for g. That is, the curve represents the substitution cost of one life history trait for another. This interpretation shows that this measure is analogous to the economic concept of the marginal elasticity of substitution. In the following, we refer to the substitution elasticity as $\psi =e_g/e_h$.

Fig. 1.

Hypothetical constraint between some (age-specific) survival and fertility rates. The iso-fitness curve represents combinations of fertility and survival that yield the same fitness. The substitution elasticity, $\psi =e_S/e_F$, represents the slope of this constraint curve at the observed rates.

Demographic Data

We apply this framework for understanding the constraints on selection for life histories to a selection of human populations. The populations are representative of the variety of human demographic experience. Livi-Bacci (17) has suggested that the variety of human demographic experience can, to a first approximation, be represented in a 2D space where the dimensions correspond to life expectancy at birth, ${\stackrel{\circ }{e}}_0$, and total fertility rate, TFR. We focus our analysis on four populations representing different combinations of high and low ${\stackrel{\circ }{e}}_0$ and TFR that bracket the demographic space described by Livi-Bacci, a strategy used in ref. 14. Two hunter−gatherer populations, the Ache and the !Kung (18, 19), are characterized by low ${\stackrel{\circ }{e}}_0$. However, the Ache have high fertility whereas the !Kung have very low fertility, with TFRs of ∼8 and 4, respectively. We use two state-level populations: Venezuela (1966) and the United States (2005) are characterized by high ${\stackrel{\circ }{e}}_0$ values. In 1966, Venezuela had very high fertility, whereas the United States is characterized by low total fertility. Table 1 summarizes the demography of the four populations along these two dimensions.

Table 1.

Four populations used in the analysis along with demographic summary statistics

Population	TFR	${\stackrel{\circ }{\mathit{e}}}_0$	T
Ache	8.09	38.5	26.3
!Kung	4.69	34.6	26.2
Venezuela	6.38	67.7	24.4
United States	2.57	74.3	28.0

T, mean age of childbearing; ${\stackrel{\circ }{\mathit{e}}}_0$, life expectancy at birth.

We will compare the results of the constraints on human life histories with two nonhuman primate species for which we have detailed data on age-specific survival and fertility. Chimpanzees (Pan troglodytes) share a most recent common ancestor with humans, making comparisons between their life histories and those of humans particularly important for understanding human evolution. We use a composite life table assembled by Hill et al. (20) from long-term field observations at a variety of chimpanzee research sites and a similar composite fertility schedule assembled by Emery Thompson et al. (21). Rhesus macaques (Macaca mulatta) are a large-bodied cercopithecoid monkey that has a demography and reproductive biology that has been extensively studied. We use demographic data from ref. 22.

An important condition for the comparability of fitness elasticities (and derived measures such as ψ) is that the age classes must be consistent across the populations/species being compared. Because of the length of human life cycles, human demographic data are typically recorded in quinquennia, whereas demographic data for many nonhuman species are recorded annually. To ensure that all age classes were equivalent for our analyses, we interpolated annual rates using smoothing splines fit to the quinquennial demographic schedules for the human populations.

Results

Fig. 2 plots the trade-off between fertility and infant survival ($P_0$). This figure shows very clearly that selection on fertility is highly constrained by the trade-off with infant survival. Even at their highest point, these curves show that fertility would need to be increased by approximately a factor of 20 to substitute for a single unit of proportional infant survival. The highest points in the curves for the Ache and !Kung lie ∼5 y below the mean age of childbearing (T), and the cost of substitution increases very rapidly for ages older and younger than T. For the United States and Venezuela, the maximum of the curves fall approximately at T (Venezuela, with its very high fertility, has the lowest T of all four populations). The curves are somewhat asymmetrical, with the costs increasing more steeply for earlier fertility. The minimum age-specific substitution elasticity for the Ache and !Kung is very similar at ∼0.045, whereas the values of the United States and Venezuela are lower, at ∼0.07 and 0.06, respectively.

Fig. 2.

Elasticity of substitution of fertility for infant survival ($P_0$) in the four human populations. For all ages, this trade-off strongly favors infant survival.

In Fig. 3, we plot the same curves for the two natural-fertility, hunter−gatherer populations along with the corresponding curves for the two nonhuman primate species. For this plot, absolute ages have been scaled by the population’s respective generation times T to ease the comparability of species with quite different life spans. Two observations immediately emerge from this plot. First, although the overall shape of the rhesus macaque curve is broadly similar to the human curves, it is shifted substantially up, meaning that the threshold for switching from investing in juvenile survival to fertility is lower for this life cycle than it is for any of the human life cycles or for chimpanzees. Second, although the maximum substitution elasticity between chimpanzees and humans are similar, the overall shape of the chimpanzee curve is quite different. The elasticities of substitution of fertility for early survival generally fall more slowly for chimpanzees than for humans, remaining quite flat throughout the middle portion of the chimpanzee life cycle.

Fig. 3.

Elasticity of substitution of fertility for infant survival ($P_0$) in the two natural-fertility, hunter−gatherer populations compared with the two nonhuman primate species.

Discussion

Our analysis has emphasized the trade-off between early (i.e., prereproductive) survival and adult fertility. In a species characterized by the intense obligate parental investment that ours is, we expect the trade-off between quantity and quality of offspring to impose severe constraints on the human life history. Indeed, using an approach to measuring life history constraints based on fitness elasticities, we show that early survival is always favored with respect to fertility. Plots of the substitution elasticities, which measure the slope of constraint curves linking life history traits in the vicinity of an equilibrium, show that the slope of the trade-off exceeds unity for all ages, indicating that an increase of fertility would need to exceed by a factor of at least 15–20 a unit proportional cost of early survival. On balance, the constraints of the human life cycle favor infant and child survival at the expense of heightened fertility. The maximum substitution elasticity is lower and occurs at an earlier maternal age in the populations with high infant mortality (Ache and !Kung) than for the two low-mortality state-level populations (Fig. 2). Although the macaque substitution elasticity curve has a similar shape to the human curves—albeit shifted up considerably—chimpanzees have a quite differently shaped curve. Although the human curves fall steeply both before and after their minima near T, the chimpanzee curve reaches its maximum much earlier and remains quite flat throughout much of the life cycle (Fig. 3). That the age-specific substitution elasticity curves fall more steeply for humans suggests an explanation for human reproductive senescence. In particular, as the marginal fitness gains of further fertility decline (23), the slope of the constraint linking fertility and infant and early juvenile survival gets too steep.

Clearly, reproduction is favored at some point and, indeed, the elasticities that we calculate are conditional on the values of all of the survival and fertility parameters in the life history. Although cross-cultural work on the costs of child rearing (24, 25) has shown that children are a net economic liability for parents as late as their early twenties among hunter−gatherers, we expect the payoffs to intensive parental investment characteristic of infancy and early childhood to show strongly diminishing marginal returns. That is, there comes a point in a child’s development when the constraint is satisfied and resuming reproduction is really worth 20 times the benefit of a comparable investment in the maturing child. Our approach thus suggests an important measurement for future empirical human life history studies.

As with the analysis of elasticities presented in ref. 14, the shape of the trade-off curves is remarkably insensitive to the specifics of the age-specific vital rates. The curves for all of the examined trade-offs have qualitatively very similar shapes. This shape will be broadly similar for any species with a late age at first breeding and iteroparous reproduction. What we expect to be dramatically different between species is the rate at which the benefits of differential investments change. For example, although the elasticity of substitution of fertility for infant survival is similar between chimpanzees and humans (Fig. 3), we expect the benefits of resumed reproduction to increase at a substantially faster rate in chimpanzees. Novel derived human developmental constraints such as substantial secondary altriciality and culturally mediated adaptations such as the existence of highly-processed complementary foods (26) mean that human infants not only can benefit longer from more intensive investment than chimpanzee infants but that mothers (and, crucially, others) have the ability to invest heavily well into childhood (27).

A key assumption in our analysis is that the life history is subject to stabilizing selection. How sensitive are the results to violations of this assumption? There are two ways that this assumption could be violated. First, the new combination of demographic parameters could be such that $\mathrm{\Delta }\lambda >0$. Second, the constraint might be relaxed, leading the left-hand side of Eq. 3 to be greater than zero. These might happen, for instance, when ecological or technological change lead to transient increase in mean fitness or a relaxation of a constraint, respectively. Suppose that the change in mean fitness following a perturbation was not zero but instead was some value δ. Equating Eqs. 3 and 2, as before, shows that the elasticity ratio will overestimate the constraint. Similarly, when the constraint is relaxed, the elasticity ratio will underestimate the constraint. Two related points suggest that this is not an important problem with the approach. First, if mean fitness is changing rapidly, then no analysis is likely to correctly predict the eventual equilibrium. Eventually, the life history will be moved to an attractor, where it will again be subject to stabilizing selection. Second, all of the life histories we have analyzed show broadly similar constraint curves. This suggests a broad basin of attraction for the human life history and that our results are robust to the assumptions of the approach.

Rather than simply representing an optimal life history, the observed demographic parameters in real populations are likely to reflect a mixture of evolutionary and mechanistic constraints as well as idiosyncratic environmental variability. For example, genetic load arising from accumulating deleterious mutations will push life histories from their optima (28). Similarly, variable environments produce phenotypes that represent compromises between those optimal for specific environmental realizations. Although our measure will not strictly capture the slope of a constraint surface at the edge of a feasible set of demographic rates, it applies to small perturbations from the complex equilibrium phenotype. As shown in ref. 28, the characteristic timescale for the clearance of deleterious mutations is on the order of thousands of generations. Our measure of the fitness effects of local perturbations matters over relatively shorter timescales, and so is relevant for adaptation if not optimization. The selection gradient in the multivariate breeders equation (29) has proven extremely useful, despite similar concerns about genetic load and fluctuating selection keeping fitness below its maximum. Lande (30) has shown that as long as environmental variability is stationary, fluctuating selection still yields a fitness surface in which the mean phenotype will follow the gradient toward the optimum. Although stabilizing selection can be swamped by stochasticity in the haploid one-locus model, the model for quantitative characters—as demographic rates surely are—produces a quasi-stationary distribution centered on the optimum phenotype under quite reasonable conditions (30).

The approach to measuring demographic constraints that we present is similar to one suggested by Caswell (16). A key difference is that our measure is done on a proportional scale and is comparable to common measures in economics. Our method also has a clear geometrical interpretation: the slope of the constraint function at its equilibrium value (Fig. 1). Caswell limited his analysis to trade-offs between current survival and later survival and between current fertility and subsequent survival. We apply our approach to a different life history question, noting that the method can be applied to any pair of demographic traits that may constrain each other. One potentially very useful application of our formalism is to investigate the constraints of lower-level variables that contribute to the elements of the demographic projection matrix. For example, the work of Hawkes et al. (31) suggest that human menopause is favored because elder women derive greater fitness benefit from provisioning their grandchildren than in producing more children of their own. It is conceivable that we could write survival and fertility in terms of women’s average production [e.g., as done in the work of Kaplan and Robson or Tuljapurkar and colleagues (3, 32, 33)]. The elasticities of substitution of women’s fertility with respect to age-specific production would provide a means of measuring the substitutability of older women’s production for fertility. For some lower-level parameter θ, the elasticity is defined by

$$e_{\theta }=\frac{\theta }{\lambda }\frac{\partial \lambda }{\partial \theta }=\frac{\theta }{\lambda }{\displaystyle \sum _{i,j}\frac{\partial \lambda }{\partial a_{ij}}\frac{\partial a_{ij}}{\partial \theta }}.$$ [5]

The logic of our analysis also applies to male life histories as much as to females. Two-sex demographic models are nonlinear, so the analyses used here do not strictly apply when both sexes are considered simultaneously. However, elasticity analysis is a useful tool for exploring the direction of selection at specified points in nonlinear models (e.g., on attractors or early on in a trajectory). The difficult part of two-sex models is using a sensible “marriage” function (34), essentially the function that maps fecundity to the abundance of females and males. We expect the substitution cost of early survival for fertility to be sensitive to this function. Schoen (34) advocates a harmonic mean marriage function for contemporary state societies with proscribed monogamy. Under such a model, we expect that males and females would experience broadly similar constraints. However, marriage functions that have their maxima under polygyny should be quite different. Although beyond the scope of this paper, the invasion exponent approach described, for example, by Benton and Grant (35) generalizes the eigensystem-based analysis we use and could be used to investigate the impact of different marriage functions. There is a frequent assumption in the literature on human mating systems that greater reproductive opportunities are necessarily better for men. Clearly, the outcome of such a hypothesis depends upon the trade-offs between heightened fertility and other life history features as well as the relative benefits of differential investment as mediated by local ecology and social constraints.

Finally, we note that specific adaptations to the steep constraint between fertility and infant survival, having arisen through human evolutionary history, could play a role in fertility transitions commonly seen following economic development and the concomitant decline in infant mortality. Although the great majority of variation in observed fertility across human populations is broadly environmental (including social), the trajectory established by life history constraints may help shape social preferences or cultural−evolutionary dynamics. As technologies and social institutions evolve to allow greater investment farther into a child’s development, it is quite plausible that the benefits of further reproduction could fall below the substitution curve of Fig. 2, favoring continued investment in children at the expense of fertility.